I am looking for an analytical expression describing the time evolution of the spatially averaged density (in practice, the average concentration in particles/m3) of a system of particles hopping with a given frequency on a d=3 lattice where some traps are randomly placed. The idea is that when a particle meets a trap it disappears from the system, so that the average density decreases to zero with some decay law. I expect to find something like a stretched exponential decay and I would like to relate the parameters of this decay curve with the input data of my system (e.g. intial particle density, traps concentration etc.) The lattice can be ordered (say, cubic) or disordered (like in glassy media or polymers) and the traps can have a finite size. As a further complication, the effect of traps saturation can be also taken into account.
Thanks!
The problem looks related to the one authors of optical-shaker were trying to look at. they looked at the concentration of atoms in a "shaking" standing wave lattice. The topic was reviewed as a method for optical cooling.
I.S. Averbukh and Y. Prior, Phys. Rev. Lett. 94, 153002 (2005)
I am not sure how far have they went with analytics, though.
Dear Yuri and Christian
Thank you very much for your answers. I think that the problem is a little different compared to your propositions. Both the shaking optical lattice (which, by the way, is not 3D but 1D) and the solution of standard diffusion equation are situations where the number of diffusing particles is conserved. My problem instead is similar to the one treated in the following paper:
Richards, Phys. Rev. Lett. 56, 1838 (1986)
where the number of particles decay to zero in some finite time. Please note that I am interested in the space-averaged, time dependent solution, i.e.
N(t) = Int (-inf; + inf; C(t,x,y,z) dx dy dz)
where N is the particle number and C is their concentration, as a function of time and space.
In particular, I suppose that the intial condition C(0, x,y,z) = constant holds, i.e. the intial concentration of hopping particles is uniform in the space. By translational invariance, also the solutions at t different from 0 are spatially uniform and therefore it is pointless (at this stage) to look for the space dependence.
Thank you again
This could potentially be seen in a different context. As an example, Feinberg described photorefractivity in terms of hopping particles (Feinberg, J; Heiman, D; Tanguay Jr, AR; Hellwarth, RW; Tanguay, JAR; and Hellwarth, RW. “Photorefractive Effects and Light-Induced Charge Migration in Barium Titanate,” J. Appl. Phys., vol. 51, no. 3, pp. 1297–1305, 1980.); Kukhtarev looked at it using semiconductor methods of analysis (Kukhtarev, N V; Markov, VB; Odulov, SG; Soskin, MS; and Vinetskii, VL. “Holographic Storage in Electrooptic Crystals. I. Steady State,” Ferroelectrics, vol. 22, pp. 949–960, 1979.). The results were the same, but the Kukhtarev method is so much easier (looking at the entire population, rather than at individual particles) that it is the one most people use today.
A lot depends on whether your traps are mobile (that is, if they too diffuse) or are fixed, but placed at random. In the former case, decay is eventually exponential in the number of distinct sites visited, S(t). S(t) itself depends on the dimension
S(t)=sqrt(t) d=1
S(t)=t/ln(t) d=2
S(t)= const. t for all higher dimensions.
If, on the other hand, the traps are immobile, then you have a stretched exponential
exp[-t^(d/(d+2))]
where d is the dimension. This arises from the fact that you might start in a region where there are exceptionally few traps, and survival is dominated by these events.
However, even in the simplest case of a point particle diffusing on a discrete lattice, there are no analytic expressions to my knowledge.
The literature is rather huge: the following somewhat random choice may get you started:
Havlin, S., Weiss, G. H., Kiefer, J. E., & Dishon, M. (1984). Exact enumeration of random walks with traps. Journal of Physics A: Mathematical and General, 17(6), L347.
Grassberger, P., & Procaccia, I. (1982). The long time properties of diffusion in a medium with static traps. The Journal of Chemical Physics, 77(12), 6281-6284.
Thank you very much Francois! I think that your answer is what I was looking for! I'll have a look on that.
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